Straight Skeleton Computational Geometry and Geometric Computing - - PowerPoint PPT Presentation

Computational Geometry and Geometric Computing Seminar Lavinia Dinu

Straight Skeleton

Supervisor: Eric Berberich

How to fit a roof to these walls?

[SSA] Walls and fitted roof

Outline

 Definitions and Basic Properties  A Simple Algorithm  Algorithm for Polygons  Applications

 Fold and Cut Problem  Roof Design

Definitions

Planar Straight Line Graph

 Embedding of a planar graph in the plane  Edges mapped to straight line segments  No self-intersections

 Figures - connected components

 Terminals - vertices of degree one  Try it yourself: www.planarity.net

[Wiki] A planar straight line graph

Skeleton

 Partition of plane into regions  Reflects the geometric shape of the graph  Arcs and nodes  Medial axis – non-linear

 Points that have more than one closest point on the boundary

[Wiki] Medial axis of a polygon

Straight Skeleton: Wavefronts

 Generate wavefronts  Vertices move on angular bisectors  Propagate wavefronts simultaneously

[SSG] Polygonal figures and initial wavefronts in red

Straight Skeleton: Events

 Edge event

 Edge length decreases to zero

 Split event

 Wavefront edge splits due to interference

Straight Skeleton: Definition

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Straight Skeleton: Example

[SAS] Blue wavefronts and red skeleton of a polygon

Straight Skeleton: Example

[SSG] Red straight skeleton of graph

Properties

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Basic Algorithm

Idea

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Initial triangulation

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Initial triangulation

Triangulation of the graph (up) and triangulation of the initial wavefronts (down) for two figures. Blue edges represent spokes, red edges wavefronts.

Flip Event

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[SAS] Flip Event

Edge Event

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[SAS] Edge Event

Split Event

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[SAS] Split Event

Complexity analysis

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A Better Algorithm

Input

 Simple non-degenerate polygon  Can have holes  Interior polygon will have the opposite orientation

[SSI] Polygon with hole

Idea & Data Structures

 Simulate the progression of the wavefronts  Use set of circular lists of active vertices – SLAV

 Each LAV – loop of vertices of a wavefront  Each vertex

 References to both neighbors  References to adjacent edges in the original polygon

 Doubly connected circular list of active vertices

 Priority queue

 events

Convex Polygons: Initialization

 Store all vertices in one LAV  For each vertex

 Add reference to the two incident edges  Compute the angle bisector

 Compute nearer intersection of bisector with adjacent ones

 Add to priority queue  Sort by distance to the corresponding edge  Add reference to the corresponding vertices  Can be duplicities among the intersection points

Convex Polygons: Initialization

[SSI] Initialization

Convex Polygons – Loop Part 1

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Convex Polygons – Loop Part 2

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Convex Polygons – Loop Part 3

[SSI] First Step

Non-convex Polygons

 Add event type attribute

 Edge event or split event

 Reflex vertices can cause both types of events

[SSI] Reflex vertices causing edge event A and split event B

Determine split events

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[SSI] Split event and feasible area

Processing split events

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[SSI] Updating the LAVs

Complexity Analysis



Applications

Applications

 Decomposition of concave polygons into convex regions

 Preprocessing step for shape matching, Minkovski sum

 Interpolating polygonal curves  Fold and cut problem  Roof design  Terrain reconstruction

Fold and Cut Problem

 Fold a piece of paper and cut along a line  Any possible shape can be obtained  Creases are:

 The straight skeleton  Perpendicular lines to the given shape reflected at the skeleton

 [PR] Applet to generate the folds

Fold and Cut Problem

[FC] Fold Pattern for a figure

Roof Design: Hip Roof

 All facets have the same slope  Algorithm:

 Construct straight skeleton  Determine distance of each vertex from supporting edge  Determine roof planes  Raise vertices accordingly

 Can be adapted for different slopes

 Change propagation speed of wavefronts

Roof Design: Hip Roof

[AGF] Steps for creating a hip roof

Roof Design: Gable Roof

 Has an small gable on top  Algorithm

 Find a vertex created by

bisectors from the original polygon

 Move it to the midpoint of the

supporting edge

[Wiki] Gable roof [AGF] Creating a gable roof

Summary

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Sources

 [SSG] O. Aichholzer, F. Aurenhammer. Straight Skeleton for General

Polygonal Figures in the Plane

 [ANT] O. Aichholzer, F. Aurenhammer, D. Alberts, B.Gaertner. A Novel

Type of Skeleton for Polygons

 [SSI] P

. Felkel, S. Obdrzalek. Straight Skeleton Implementation

 [AGF] R. G. Laycock, A. M. Day. Automatically Generating Roof Models

from Building Footprints

 [SAS] D. Eppstein. A Subquadratic Algorithm for the Straight Skeleton  [FC] http://erikdemaine.org/foldcut/  [PR]

http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/EricBiunno/

 [Wiki]Wikipedia

Thank you for your attention!

Questions?